The 2-higher and 3-higher Mahler measure of some rational functions are given in terms
of special values of the Riemann zeta function, a Dirichlet L-function and multiple polylogarithms. Our results generalize those obtained in [10] for the classical Mahler measure.
We improve one of our results by providing a reduction for a certain linear combination
of multiple polylogarithms in terms of Dirichlet L-functions. We conclude by
giving a complete reduction of a special case.